2021/06/09 by Ivan Tomašić, Tomasic, Ivan, Michael Wibmer +1
Computer Science · Mathematics · #12H05 #18D40 #18E50 #37B10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Category Theory (math.CT) #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2106.05334
openalex publication_date 2021/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a Galois theory for difference ring extensions, inspired by Magid's separable Galois theory for ring extensions and by Janelidze's categorical Galois theory. Our difference Galois theorem states that the category of difference ring extensions split by a chosen Galois difference ring extension is classified by actions of the associated difference profinite Galois groupoid. In particular, difference locally étale extensions of a difference ring are classified by its difference profinite fundamental groupoid. The emergence of difference profinite spaces, viewed as discrete dynamical systems in the realm of topological dynamics, leads us to investigate the interaction of difference algebra and symbolic dynamics. As an application of this interaction, we prove the near-rationality of a certain difference zeta function counting solutions of systems of difference algebraic equations over algebraic closures of finite fields with Frobenius.